If G is a DAG we denote by τ (G) the number of distinct topological sorts of G. Now let G = (V, E) be any DAG. An add-one-vertex operation on G constructs a digraph G 0 = (V 0 , E 0 ) such that • G 0 is also a DAG. • We add exactly one new vertex, call it z that is, V 0 = V ∪ {z}, z ∈ / V . • We add some (possibly 0) new directed edges, that is, E 0 ⊇ E. (a) Prove that for any DAG G = (V, E) there is a way to perform an add-one-vertex operation on G such that the resulting DAG G 0 has the same number of topological sorts, i. e. τ (G 0 ) = τ (G). (b) Prove that for any DAG G there is a way to perform an add-one-

At river bank, checking account customers earn 0.75% interest a month on their account balances, pay no service fees for writing checks, and pay a monthly $2.00 financial charge. calculate the income earned in one month on the checking account for a customer with an account balance of $1,000. a. $4.25 b. $5.50 c. $6.75 d. $7.25 checking account earnings at baker’s bank are expressed by the equation: i = -0.09x + 10.2. while the earnings at elite bank are modeled by: i = -0.02x + 7.5. in both cases, x is the number of checks written. at what number of checks will elite bank start generating more checking account income than baker's bank? a. 12 b. 25 c. 39 d. 42 emilio’s checking account had a balance of 728.32 at the beginning of the week. he wrote checks for the following amounts throughout the week: $99.48, $33.50, $18.23, and $72.05. he also deposited his paycheck in the amount of $1,109.90. what is his account balance at the end of the week? a. $1,192.35 b. $1,222.86 c. $1,540.52 d. $1,614.96